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Principle Bundles and Hopf Map for Spheres

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Principle Bundles and Hopf Map for Spheres Principle bundles and Hopf map for spheres Irina Markina, University of Bergen, Norway Summer school Analysis - with Applications to Mathematical Physics Gottingen¨ August 29 - September 2, 2011 Principle bundles and Hopf map for spheres – p. 1/46 Fiber bundle The smooth fiber bundle is a collection of smooth manifolds F,M,B and a smooth map π : M → B. ∀ x ∈ M, there is U(π(x)) ⊂ B, such that π−1(U) is diffeomorphic to the product space U × F : φ M ⊃ π−1(U) / U × F, oo ooo π oo oo pr woo 1 B ⊃ U ∀ b ∈ B, the pre-image π−1(b) is diffeomorphic to F and is called the fiber over b. B is the base space, M is the total space and the map π is the projection map. Principle bundles and Hopf map for spheres – p. 2/46 Fiber bundle Sea urchin Principle bundles and Hopf map for spheres – p. 3/46 Vertical space Fiber bundle π : M → B dqπ : TqM → Tπ(q)B is surjective ⇒ ker(dqπ) = ∅ Denote ker(dqπ) = Vq and call the vertical space. The collection of all vertical space is vertical distribution or vertical sub-bundle V ⊂ TM. Vq = Tq(F ) for fiber F passing through q Principle bundles and Hopf map for spheres – p. 4/46 Ehresmann connection An Ehresmann connection for π : M → B is a distribution D ⊂ TM that is everywhere transverse and of complementary dimension to V : TqM = Dq ⊕ Vq. q ∈ M. ⊕ denotes only transversality, but not orthogonality. Dp is a horizontal vector space and the Ehresmann connection D is a horizontal distribution on M. Given π : M → B we have V = ker(dπ). Construction of D is the question of a mathematical art. Principle bundles and Hopf map for spheres – p. 5/46 Sub-Riemannian structure by restriction Let π : (M,gM ) → B, Vq = ker(dqπ) Define Dq ⊕⊥ Vq = TqM. The distribution D is the Ehresmann connection Let gD := gM |D (M,D,gD) is the sub-Riemannian manifold defined by submersion π and the Riemannian metric gM . The sub-Riemannian length of a horizontal curve is equal to the Riemannian length, since the vertical components vanish. Principle bundles and Hopf map for spheres – p. 6/46 Induced sub-Riemannian structure Let π : M → (B,gB) and D is defined. Pullback the metric gB to the horizontal distribution D: gD(v,w) := gB(dqπ(v),dqπ(w)), v,w ∈ Dq, q ∈ M. (M,D,gD) is sub-Riemannian manifold induced by the Ehresmann connection D and the Riemannian metric gB. If γ is horizontal in M, then lsR(γ) = lgB (π(γ)). The vertical component of γ˙ is absent Principle bundles and Hopf map for spheres – p. 7/46 Induced sub-Riemannian structure The horizontal lifting of c: I → B is a curve γ : I → M such that γ˙ (t) ∈ Dγ(t) and π(γ(t)) = c(t) for all t ∈ I. The horizontal lift of a Riemannian geodesic in B is a sub-Riemannian geodesic in M. If the Riemannian geodesic in B is a length minimizers between b0,b1 ∈ B, then its horizontal lift is a sub-Riemannian length minimizer between the corresponding fibers. Principle bundles and Hopf map for spheres – p. 8/46 When two structures coincide? Suppose π : (M,gM ) → (B,gB), gMD := gM |D. and D is given. If the restriction dqπ|Dq : (Dq,gMD ) → (Tπ(q)B,gB) is a linear isometry for all q ∈ M, then (M,D,gMD )=(M,D,gBD ). The submersion π is called the Riemannian submersion. Principle bundles and Hopf map for spheres – p. 9/46 Principal bundle A smooth fiber bundle (F,M,B,π) is a principal bundle if a fiber F has the structure of a Lie group G and µ: F × G → F q.τ → rτ (q) = qτ acts (on the right) freely and transitively on each fiber. G acts freely on the right on M if ∀ q ∈ M, q.τ = q.ς if and only if τ = ς. G acts transitively on the right on M if ∀ q,p ∈ M there is τ ∈ G such that q.τ = p Principle bundles and Hopf map for spheres – p. 10/46 Metric on principal bundles Let π : (M,gM ) → B, Vq = ker(dqπ), D ⊕⊥ V and let gM be G-right invariant: gM (vq,wq) = gM drτ (vq), drτ (wq) , τ ∈ G Then gD = gM |D is the sub-Riemannian metric gV = gM |V is the vertical metric Fibers are isomorphic to a Lie groups G, so gV can be encode in the Lie algebra g. Principle bundles and Hopf map for spheres – p. 11/46 Inertia tensor The infinitesimal generator σq of the right G-action on M d σq : g → Vq ⊂ TqM such that g ∋ ξ → q. exp(ǫξ). dǫǫ=0 d d σq(ξ) = q. exp(ǫξ) = lq exp(ǫξ) = dlq(ξ) dǫǫ=0 dǫǫ=0 Define the bilinear symmetric tensor Iq by Iq(ξ,η) = gV (σq(ξ),σq(η)), ξ,η ∈ g q ∈ M. Iq is the inertia tensor on g. If Iq is independent on q, then gV is also left invariant. Principle bundles and Hopf map for spheres – p. 12/46 Metric we are working with Let π : (M,gM ) → B, Vq = ker(dqπ), D ⊕⊥ V A Riemannian metric gM is of constant bi-invariant type if 1. gM is right G invariant, 2. its inertia tensor Iq is independent of q ∈ M. Metric we are working with Let π : (M,gM ) → B, Vq = ker(dqπ), D ⊕⊥ V A Riemannian metric gM is of constant bi-invariant type if 1. gM is right G invariant, 2. its inertia tensor Iq is independent of q ∈ M. A σq . y . o proj. g l =∼ ker(dqπ) = Vq o TqM −1 σq A is g-valued connection form Principle bundles and Hopf map for spheres – p. 13/46 Normal geodesics Let expR be the Riemannian exponential map for gM , γv,R(t) = expR(tv) be the Riemannian geodesic passing through q ∈ M with the initial velocity vector v ∈ TqM. Lift horizontaly π(γv,R) from B to M and get a curve γsR. The curve γsR is a normal sub-Riemannian geodesic, given by γsR(t) = γv,R(t)expG(−tA(v)). All normal sub-Riemannian geodesics are given by this formula Principle bundles and Hopf map for spheres – p. 14/46 Proof of the theorem gM , gD, gV =⇒ HR, HsR, HV expR : TM → M, expsR : D → M, expV : V → M. If Hamiltonian functions Poisson commute, then the flows on T ∗M also commute. If v = vD + vV is the initial velocity vector and the exponential maps commute expsR(tvD) = expR(tv)expV (−tvV ). The last step is to observe expV (−tvV ) = expG(−tA(vV )) since the metric gV is bi-invariant. Principle bundles and Hopf map for spheres – p. 15/46 Proof of the theorem ∗ ∗ {HV ,HsR} = 0, ((τ,µ) × (b,p)) ∈ T G × T U, U ∈ B HsR = HsR(µ,b,p), HV = HV (µ) l k ∂HsR ∂HV ∂HsR ∂HV {HsR,HV } = + ∂µj ∂τj ∂pj ∂bj j=1 j=1 l k ∂H ∂H ∂H ∂H − sR V + sR V = 0. ∂τj ∂µj ∂bj ∂pj j=1 j=1 Principle bundles and Hopf map for spheres – p. 16/46 Hopf map Let S2n+1 = {z ∈ Cn+1 | z2 = 1} right U(1)-action on S2n+1 is (z0,...,zn).υ =(z0υ,...,znυ), υ ∈ U(1) It induces the principal U(1)-bundle h U(1) → S2n+1 −→ CP n (z0,...,zn) → h(z0,...,zn)=[z0 : · · · : zn], where [z0 : · · · : zn] denotes homogeneous coordinates. j j zj = x + iy Principle bundles and Hopf map for spheres – p. 17/46 Odd dimensional spheres Any sphere S2n+1 possesses the vector field 0 0 n n V˜q = −y ∂x0 + x ∂y0 − ... − y ∂xn + x ∂yn . n+1 that is V˜q = iNq, where i is the usual i in C and 2n+1 n+1 Nq = q is normal to sphere S ⊂ C . V˜q ∈ ker(dqπ), Vq = span{V˜q}, D ⊕⊥ V, It is also D = ker ω for the contact form 0 0 n n ω = −y0dx + x0dy − ... − yndx + xndy Principle bundles and Hopf map for spheres – p. 18/46 Geodesics on S2n+1 2n+1 2n+1 S =(S ,D,gD) is the sub-Riemannian manifold The Lie algebra u(1) = spanR(i) The u(1)−valued connection form is A(v) = ig(v,Vq), 2n+1 v ∈ TqS The Riemannian metric g on S2n+1 is of constant bi-invariant type, since for iα ∈ u(1) d σq(iα) = q exp (ǫ(iα)) = q.iα = αVq, dǫ u(1) ǫ=0 Iq(iα, iα˜) = g(αVq, αV˜ q) = αα˜ does not depend on q ∈ M Principle bundles and Hopf map for spheres – p. 19/46 Geodesics on S2n+1 2n+1 2n+1 Let q ∈ S and v ∈ TqS . If v γ (t)=(z (t),...,z (t)) = q cos(vt) + sin(vt) R 0 n v where v = g(v,v), is the great circle satisfying γR(0) = q and γ˙R(0) = v, then the sub-Riemannian geodesic γsR is given by −itg(v,Vq) −itg(v,Vq) γsR(t) = z0(t)e ,...,zn(t)e , t ∈ R. Principle bundles and Hopf map for spheres – p. 20/46 Corollaries 2 2 2 • γ˙sR(t) + g (v,Vq) = v . γ˙sR is constant and length of γ is 2 2 ℓsR(γ)=(b − a) v − g(v,Vq) for t ∈ [a, b] . Principle bundles and Hopf map for spheres – p. 21/46 Corollaries 2 2 2 • γ˙sR(t) + g (v,Vq) = v .
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